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Higher order Poincaré-Pontryagin functions and iterated path integrals. (English) Zbl 1104.34024

Consider a nonintegrable foliation \(df-\varepsilon(Pdx+Qdy)=0\), where \(\varepsilon\) is a small parameter and \(f,P,Q\) are polynomials in the two variables \(x,y\). Assume that the unperturbed system \(df=0\) has an annulus of periodic trajectories \(\{\gamma(t)\}\), where \(\gamma(t)\subset\{f=t\}\) for \(t\in (a,b)\). The related first return map has the form \[ {\mathcal P}(t,\varepsilon)=t+\varepsilon^k M_k(t) +\varepsilon^{k+1} M_{k+1}(t)+\ldots, \] where \(k\) is a natural number. The zeroes of the generating function \(M_k\) in \((a,b)\) correspond to the limit cycles produced by the period annulus after the perturbation.
Here, the author takes an analytic continuation of the generating function and establishes several facts about it. It is proved that \(M_k(t)\) is a linear combination of iterated path integrals along \(\gamma(t)\) of length at most \(k\) over certain rational one-forms. \(M_k\) is a function of moderate growth and its monodromy representation is finite-dimensional. Moreover, \(M_k\) satisfies a linear differential equation of Fuchs type and its monodromy group is contained in \(\text{SL}(n,\mathbb{Z})\), where \(n\) is the order of the equation. One has \(n\leq r^k\), where \(r=\dim H_1(f^{-1}(t_0),{\mathbb Z})\) and \(t_0\) is a typical value of \(f\).

MSC:

34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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